4
3
2
1
0
;1
;2
;3
;4
;5
;10
;8
;6
;4
;2
0
2
4
6
8
10
The subset of slice that has an I/O step response sensi-
Figure
11.15
H
tivity magnitude, at = 1, of less than 0 75. This is the set of points for
t
:
which (11.13) holds.
11.5
Robustness Specifications
11.5.1
Gain Margin
We now consider the gain margin speci cation +4 3:5db gm: the system should
D
;
remain stable for gain changes in Pstd
0 between +4dB and 3:5dB. In section 10.4.4
;
we used the small gain theorem to show that
T
1:71
(11.14)
k
k
1
S
2:02
(11.15)
k
k
1
are (di erent) inner approximations of the gain margin speci cation +4 3:5db gm.
D
;
Figure 11.16 shows the subset of slice that meets the gain margin speci cation
H
+4 3:5db gm, together with the two inner approximations (11.14{11.15), i.e.,
D
;
H(a)
13 + H(b)
13 + (1
)H(c)
13
1:71
(11.16)
;
;
1
1
H(a)
13 + H(b)
13 + (1
)H(c)
13
2:02:
(11.17)
;
;
;
1
In general, the speci cation +4 3:5db gm is not convex, even though in this case
D
;
the subset of slice that satis es +4 3:5db gm is convex. The two inner approx-
H
D
;
imations (11.16{11.17) are norm-bounds on H, and are therefore convex (see sec-
11.5 ROBUSTNESS SPECIFICATIONS
263
tion 6.3.2). The two approximations (11.16{11.17) are convex subsets of the exact
region that satis es +4 3:5db gm.
D
;
1:5
1
13
1 71
kH
k
:
1
;
;
0:5
0
;0:5
;1
@
I
@
1
13
2 02
k
;
H
k
:
1
@
I
@
;1:5
exact
;2
;10
;8
;6
;4
;2
0
2
4
6
8
10
The boundary of the region where the gain margin speci-
Figure
11.16
cation +4 3 5db gm is met is shown, together with the boundaries of the
D
;
:
two inner approximations (11.16{11.17). In this case the exact region turns
out to be convex, but this is not generally so.
The bound on the sensitivity transfer function magnitude given by (10.69) is an
inner approximation to each gain margin speci cation. Figure 11.17 shows the level
curves of the peak magnitude of the sensitivity transfer function, i.e.,
'max sens(
) = 1
H(a)
13 + H(b)
13 + (1
)H(c)
13
:
(11.18)
;
;
;
1
In section 6.3.2 we showed that a function of the form (11.18) is convex the level
curves in gure 11.17 do indeed bound convex subsets of slice.
H
11.5.2
Generalized Gain Margin
We now consider a tighter gain margin speci cation than +4 3:5db gm: for plant
D
;
gain changes between +4dB and 3:5dB the stability degree exceeds 0:2, i.e., the
;
closed-loop system poles have real parts less than 0:2. In section 10.4.4 we used
;
the small gain theorem to show that
S 0:2 2:02
(11.19)
k
k
1
is an inner approximation of this generalized gain margin speci cation.
264